Nonlinear Nondestructive Testing and Imaging Methods
In the last few years, a strong interest for nondestructive testing methods based on nonlinear elastic effects in solid has grown, driven by the request from industry for sensitive quantification and localization of micro-structural damage. Researchers have developed innovative techniques that explicitly interrogate the material’s micromechanical behavior and its effect on wave propagation by investigating the amplitude dependence of macroscopically observable properties , , , , , . Such techniques are termed Nonlinear Elastic Wave Spectroscopy (NEWS) techniques. The basis of all NEWS techniques is to measure and analyze macroscopic signatures resulting from a local violation of the linear stress-strain relation at the micro-scale.
Several NEWS techniques have been developed to probe the existence of damage induced nonlinearity. One of the most frequently studied methods is harmonic analysis in the frequency domain , , consisting of measuring the second and higher harmonic amplitude versus the strain amplitude of the fundamental, which provides quantitative information about the nature of the nonlinearity. Another technique consists of parametric interactions between waves emitted in materials. The sample acts as a nonlinear frequency mixer, so that sum and difference frequency waves are created , . In addition, modulations of amplitude  and phase  have been investigated in order to evaluate the classical nonlinear perturbation coefficient coming from the Taylor expansion of stress-strain relation . The study of resonance frequency provides key information about nonlinear behavior. By plotting the frequency shift as a function of the fundamental mode strain amplitude, it is observed that resonance amplitude distortion increases significantly with defect , . Tests performed on a wide variety of materials subjected to different micro-damage mechanisms of mechanical, chemical and thermal origin, have shown that the sensitivity of such nonlinear methods to the detection of micro-scale features is far greater than that obtained with linear acoustical methods .
Linear and Nonlinear Ultrasonic Imaging Methods for NDT
The most frequently used imaging techniques based on the analysis of ultrasonic signal generation and propagation are surface-scan imaging involving laser vibrometry, air-coupled ultrasonics and ultrasound thermography, ultrasound tomography; and time reversal (or wave phase conjugation) techniques. Laser vibrometry is a precise technology for non-contact vibration measurements, modal analysis and non-destructive testing for many areas of engineering. Linear laser vibrometry provides an extreme sensitivity (pico-meter range) in measuring and imaging vibration fields by evaluating the laser light scattered back from the vibrating object. Air-Coupled Ultrasound is another established method for remote defect imaging that has become a routine inspection technique in nondestructive testing for a wide range of materials and components . A new generation of air-coupled ultrasonic transducers covering a wide frequency range up to few MHz enables to image faint acoustic fields scattered by tiny defects.
In addition to pure acoustic or ultrasonic scanning measurements, techniques based on thermal-acoustic interaction have been successfully used in several NDT configurations: SPATE  is a well known method for mechanical stress measurement under static loads. With higher frequency and efficient excitation, it is also possible to visualize the dissipated energy that is released due to nonlinear effects. This technique is known as Ultrasound-Excited Thermography (ULT). Basic investigations  showed that the main advantage of ULT is the defect selective indication of material flaws by using elastic nonlinearity in defective material areas caused by friction or locally increased dissipation.
Multi-elements transducers are commonly used in ultrasonic medical imaging. They offer a great flexibility in the realisation of images, and advanced imaging techniques have been developed such as coded imaging, and dynamical focusing in reception. Most of these methods are now starting to be transferred to linear ultrasonic NDT applications, with the development of dedicated electronic systems (MultiX of the French society M2M).
In aeronautical industry, the main structures are plate like components of large size often made in composite a highly attenuating medium. When imaging these large structures for structural health monitoring, particular focus has been made on the use of guided Lamb waves , , . Generally one tris to generate a single Lamb mode to simplify the measurement and understand how such wave is scattered by various linear defects. In this case the arrays are necessarily sparse with much interest in reducing the spatial density of the sensors in order to produce a cost effective system.
Unfortunately, these traditional NDT techniques are often not sufficiently sensitive to the presence of incipient and progressive damage. Indeed, the main difficulty in the characterization of a degradation process in structural materials relates to the fact that the material exhibits very few measurable signs of damage prior to the onset of delaminations or macro-cracks. To overcome this limited sensitivity of linear imaging methods, NEWS methods have recently been extended to visualization techniques for imaging defect’s nonlinearity distributions using laser vibrometry , airborne ultrasound , shearography  or all optical photothermal and photoacoustic methods . In the airborne ultrasound method, the defects, acting as localized sources of nonlinear vibrations, efficiently radiate higher harmonics into the surrounding air.
TR and NEWS Combined Methods
Time Reversal (TR) - is now a well known technique which have been developed in different fields including medical therapy, diagnostic, and underwater acoustics, due to its ability to provide spatial and temporal focusing of an ultrasonic wave. Time-reversal invariance in acoustics means that for every burst of sound s(r,t) emitted from a source, and which is reflected, refracted, or scattered by heterogeneities of the propagation medium, here exists a set of waves s(r,t) that precisely retrace all these complexes paths and converge at the original source, as if time were going backwards. This invariance is satisfied by the equation in non attenuating media. The TR process leads to a spatial focusing and a temporal compression. Spatial focusing means that the time-reversed field focuses back exactly at the source. Temporal compression means that the time reversed signal at the source is similar to the signal previously emitted by the source. In other words, the result of a TR process is that waves recorded on the boundary are focused back in space and time on the acoustic source, or on the scattering targets inside the region that were acting as sources. For classical linear TR process, the returned signal focuses on the direct wave source position not on the defect , . The size of the focal spot depends on source size and form, and on the frequency of the signal emitted. Concerning NDT applications, TR processes have been applied in several classical ultrasonic inspection methods: C-scan with immersed samples , Rayleigh and Lamb waves propagation in plates and hollow cylinders , , , and structural health monitoring , . In these studies, it was shown that the TR principle improves the detection of flaws in heterogeneous materials for which the microstructure displays a strong speckle noise that is obstructing the observation of a defect echo in classical ultrasonic inspection. On the other hand, researchers have encountered a serious limitation of the traditional TR technique in the fact that only the strongest scatterer can be imaged. The application of the so-called Décomposition de l’Opérateur de Retournement Temporel (DORT) method ,  and successive TR iterations ,  may overcome this feature to some extent and may enhance the detection by focusing selectively on weaker scatterers. Using these advanced analysis and signal processing techniques, flaws with sizes even smaller than the wavelength can be detected in highly heterogeneous materials such as titanium alloys , .
Experiment with NEWS techniques have demonstrated that micro-damage is first of all a process of nonlinear scattering giving rise to the creation of higher harmonics, rather than to linear scattering effects. So, from this point of view, the classical TR procedure should be modified in such a way that the main signal treatment is concentrated on the nonlinear components of the signals.
Following the laboratory studies of the NEWS techniques, we can underline two important principles : (1) the macroscopically observed nonlinear signatures originate from zones with micro-damage and micromechanical nonlinear stress-strain relations; (2) the nonlinear signatures are most efficiently generated at those locations where the strain within the sample is prevailing. These two principles can be used as the basis for new micro-damage visualization techniques based on nonlinear material properties. The NEWS methods allow characterization of the nonlinear behavior, but they do not provide information about defect localization. To overcome this problem, a method combining a Time Reversal (TR) process and a nonlinear treatment has been proposed , .
For combining the nonlinearity based TR process with the NEWS methods, two technologies have been proposed, depending on whether nonlinear treatment is performed before or after the TR process. As presented in Fig 1.1, these two methodologies are defined as TR-NEWS, with nonlinear analysis as a post-treatment of time reversal, and as NEWS-TR, with nonlinear analysis as a pre-treatment of time reversal .
The TR-NEWS method, which consists in increasing locally the stress field using properties of linear TR and subsequently applying nonlinear analysis, has been experimentally demonstrated by Sutin et al. . It seems to have a wide potential for application in solid ultrasound imaging for nondestructive testing , . For TR-NEWS technology, different experimental set-ups have been recently proposed , , , , . In these experiments, generally, two high frequency signals are used to excite the medium. Then, an analysis of the intermodulation of the retro-focalized signals point by point on the imaged area is made. In the experiment of Le Bas et al. , a 1MHz signal (f1) is first sent to a first source, and the out of plane particle velocity is recorded at a chosen location using a laser vibrometer. A second signal with a 200 kHz frequency (f2) is sent at a second source and again a laser vibrometer records the signal at the same position. Both recorded signals are then time reversed and reemitted from their corresponding original transducer at exactly the same time. Doing so, the time reversal principle makes sure that both signals arrive at the same time at the fixed point where the laser picks up the out of plane vibration. The intermodulation at the focused signal in time is then analyzed in terms of the sum (f1+f2) and difference (f1-f2) spectrum components. This procedure is repeated for all points on a line crossing the flaw position. For an intact location the level of intermodulation is quite low. However, for a micro-damaged zone the intermodulation becomes very high. The nonlinearity signatures contained in the sum and difference frequencies have been obtained as function of the distance to the crack. At the position of the crack, the intermodulation signature is evidently much larger than elsewhere. A contrast about a factor of 10 was obtained.
Our TR-NEWS experiment has been realized on a fatigue steel sample combining a “chaotic cavity transducer” and a PI filtering meth od.
The other alternative to classical TR, called NEWS-TR, consist in selecting only the nonlinear or harmonic energy contained in the response signals and returning merely this part back into the medium by the time reversal process. Doing so, the time reversed signal will focus on the micro-damaged area, which is where the harmonics were created, while linear scatters will not show up at all , . This method has been described for the first time by Bou Matar et al.  and has only been validated experimentally recently . The nonlinear TR process has recently been demonstrated to be highly valuable for ultrasound imaging of damaging in solid , , , . Moreover, similar ideas have already been used in fluids where Wave Phase Conjugation (WPC) in nonlinear regime has been demonstrated for nonlinear ultrasonic imaging , . WPC is known as the spectral representation of TR transformation. The WPC technique, which originated in the field of nonlinear optics, has been adapted and applied for ultrasonic research applications in the 1980’s by the scientific group of the Wave Research Center of the General Physics Institute of the Russian Academy of Sciences . The original parametric method for acoustic WPC producing a giant (>80 dB) amplification was elaborated for the first time by this group . The advantage of the parametric WPC technique is its capability, by principle, to use a single element time reversal mirror , .
In the NEWS-TR technology, two filtering methods have been investigated to return only the nonlinear parts (harmonics) of the received signal, i.e., harmonic filtering and pulse inversion (PI) , . For the harmonic filtering, one option consists of selecting only the nonlinear or harmonic energy contained in the response signals and returning only this part back into the medium by the time reversal process. Pulse inversion is an alternative filtering procedure based on the fact that the phase inversion of a pulsed excitation signal (180o phase shift) will lead to the exact inverted phase signal within a linear medium . But, this is not the case in a nonlinear (or micro-damaged) material due to the generation of harmonics. Advantage of this information is taken by adding the response from two phase-inverted pulses (positive and negative) and sending back the sum to the receivers.
A numerical study of the comparison of the two proposed filtering methods for NEWS-TR technique, used for detecting defects with a nonlinear hysteretic behavior, has been conducted in 2D  and 3D . Hysteretic nonlinearity exhibiting high level of odd harmonics, the third harmonic signal is extracted in these numerical simulations. The results show that the higher the frequency, the greater the increase in retro-focusing quality and decreasing the source size reduces the retro-focusing quality. The simulation results demonstrate that the main difference between these two methods of filtering (harmonic filtering and pulse inversion) are: (1) Pulse inversion filtering is better for the defect detection near the edge of the sample, all information related to the linear propagation in the medium is eliminated with pulse inversion filtering contrary to that with harmonic filtering; (2) Harmonics filtering is more precise than pulse inversion filtering when the defect is located between the emitter and receiver, the higher the harmonics frequency, the smaller the retro-focusing spot size will be. An experiment of NEWS-TR with a pulse inversion filtering has been presented by Le Bas et al. . A not perfect, but encouraging result has been obtained with a one channel time reversal process in a PMMA glass material.
Table of contents :
CHAPTER 1: INTRODUCTION TO NONLINEAR NONDESTRUCTIVE TESTING AND IMAGING
1.2 Nonlinear Nondestructive Testing and Imaging Methods
1.2.1 NEWS Methods
1.2.2 Linear and Nonlinear Ultrasonic Imaging Methods for NDT
1.2.3 TR and NEWS Combined Methods
1.3 Nonlinear Elasticity and Elastodynamic Equations
1.3.1 Nonlinear 1D Propagation Model in Heterogeneous Elastic Media
1.3.2 “Classical” and “Non-classical” Nonlinear Elasticity
1.3.3 Nonlinear Elastodynamic System of Equations
1.4 Numerical Simulation Methods
1.4.1 Finite Difference Method
1.4.2 Finite Volume Method
1.4.3 Finite Element Method
1.4.4 Pseudo-Spectral Method
1.4.5 Discontinuous Galerkin Finite Element Method
1.5 Pseudo-Spectral Simulation of 1D Nonlinear Propagation in Elastic Media
1.5.1 The Elastic Wave Solver
1.5.2 Shock Wave Simulation
1.5.3 Rod Resonance Simulation
CHAPTER 2: THE NODAL DISCONTINUOUS GALERKIN METHOD
2.2 Discontinuous Galerkin Finite Element Method Scheme in 2D
2.2.1 General Formulation of Discontinuous Galerkin Schemes
2.2.2 Defining Discontinuous Galerkin Operators on Triangular Elements
2.2.3 Numerical Fluxes in the Discontinuous Galerkin Method
2.2.4 Discontinuous Galerkin Operators on Quadrilateral Element
2.2.5 Time-Stepping and Discrete Stability
2.3 Boundary Conditions
2.3.1 Open Boundaries
2.3.2 Stress Free and Fixed Surface Boundaries
2.5 Numerical Validation: Comparison with Analytical Solutions
2.5.1 Linear Isotropic Simulation of Lamb’s Problem
2.5.2 Linear Simulation of Elastic Waves Propagation in Anisotropic Apatite Material
2.5.4 Simulation of Wave Propagation in “Classical” Nonlinear Elastodynamic Material
CHAPTER 3: PML ABSORBING BOUNDARY CONDITION
3.2 C-PML for Second-Order Elastodynamic Wave Equations
3.2.1 Wave Equations for Anisotropic Solid in 2D
3.2.2 C-PML Elastic Wave Equations in Frequency Domain
3.2.3 Interpretation of C-PML as an Anisotropic Solid Medium
3.2.4 C-PML Elastic Wave Equations in Time Domain
3.2.5 Numerical Simulations
3.3 C-PML Formulation for Piezoelectric Solid
3.3.1 Wave Equations for Piezoelectric Solid in 2D
3.3.2 Formulation of C-PML in Frequency Domain
3.3.3 Formulation of C-PML in Time Domain
3.3.4 Numerical Simulations
3.4 Nearly Perfectly Matched Layer (NPML) for Elastic Solid
3.4.1 Formulation of NPML for Elastic Wave Propagation
3.4.2 Comparison of NPML with C-PML
3.5 Stabilized Absorbing Boundary Layer
3.5.1 Formulation of Stabilized Absorbing Boundary Layer
3.5.2 Stability Analysis
3.5.3 Numerical Simulations of MPML for Anisotropic Solid Medium
3.5.4 Application to Propagation in Isotropic and Piezoelectric Plate
CHAPTER 4: APPLICATION OF CHAOTIC CAVITY TRANSDUCER TO LINEAR AND NONLINEAR ELASTIC IMAGING
4.2 Principle of Chaotic Cavity Transducer
4.2.1 Principle of One Channel Time Reversal Acoustic
4.2.2 An Instructive Experiment
4.2.3 Chaotic Cavity Transducer
4.2.4 Signal Processing Methodology
4.2.5 Numerical Simulations of Chaotic Cavity Transducer
4.3 Experiments in a Reverberant Medium
4.3.1 Set-up of the Experiment
4.3.2 Experimental Results
4.3.3 Contrast of the Retro-Focalized Signal
4.4 Experiments on a Non-Reverberant Medium
4.5 Nonlinear Acoustic Imaging with Chaotic Cavity Transducer
4.5.1 TR-NEWS Experiment with Chaotic Cavity Transducer
4.5.2 NEWS-TR Experiment with Chaotic Cavity Transducer
APPENDIX A: ANALYTICAL SOLUTION FOR THE PROPAGATION OF ELASTIC WAVES IN UNBOUNDED ANISOTROPIC SOLID
APPENDIX B: C-PML MEMORY VARIABLES EVOLUTION EQUATIONS